The static dipole polarizabilities of the positively charged alkali atoms from Li+ to 119(+) (eka-Fr) were obtained from relativistic coupled-cluster theory using a scalar relativistic Douglas-Kroll Hamiltonian. Spin-orbit coupling effects were obtained from a fully relativistic four-component Dirac-Hartree-Fock scheme followed by a second-order many-body perturbation treatment to account for electron correlation. Electron correlation effects are found to be small for all ions, but become more sizable as the nuclear charge increases. Scalar relativistic effects dominate over electron correlation for Cs+, Fr+, and 119(+). Spin-orbit coupling is non-negligible for the heaviest elements Fr+ and 119(+), where they dominate over both electron correlation and scalar relativistic effects. Breit interactions obtained for Cs+ and Fr+ can safely be neglected. A relationship between dipole polarizabilities and second ionization potentials is established. The use of a basis set limit one-electron description and a high level treatment of electron correlation and relativistic effects makes our results the most accurate available for the stable dipole polarizabilities (1.00 +/-0.04 a.u. for Na+, 5.52 +/-0.04 a.u. for K+, 9.11 +/-0.04 a.u. for Rb+, 15.8 +/-0.1 a.u. for Cs+, 20.4 +/-0.2 a.u. for Fr+, and 32 +/-1 a.u. for 119(+)).